THE WILLIAMS-BJERKNES MODEL ON REGULAR TREES

• 成果类型: Article
• 署名作者: Louidor, Oren; Tessler, Ran; Vandenberg-Rodes, Alexander
• 署名单位: Technion Israel Institute of Technology; Hebrew University of Jerusalem; University of California System; University of California Irvine
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP966
• 发表日期: 2014
• 页码: 1889-1917
• 关键词: valued markov-processes; tumor-growth model; contact process; complete convergence; intermediate phase; set; Duality

摘要

We consider the Williams-Bjerknes model, also known as the biased voter model on the d-regular tree T-d, where d >= 3. Starting from an initial of healthy and infected vertices, infected vertices infect their neighbors at Poisson rate lambda >= 1, while healthy vertices heal their neighbors at Poisson rate 1. All vertices act independently. It is well known that starting from a configuration with a positive but finite number of infected vertices, infected vertices will continue to exist at all time with positive probability if and only if lambda > 1. We show that there exists a threshold lambda(c) is an element of (1, infinity) such that if lambda > lambda(c) then in the above setting with positive probability, all vertices will become eventually infected forever, while if lambda < lambda(c), all vertices will become eventually healthy with probability 1. In particular, this yields a complete convergence theorem for the model and its dual, a certain branching coalescing random walk on T-d-above lambda(c). We also treat the case of initial configurations chosen according to a distribution which is invariant or ergodic with respect to the group of automorphisms of T-d.